Table Of ContentLecture Notes in Mathematics 2154
Michèle Loday-Richaud
Divergent Series,
Summability and
Resurgence II
Simple and Multiple Summability
Lecture Notes in Mathematics 2154
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
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SylviaSerfaty,ParisandNew York
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AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Michèle Loday-Richaud
Divergent Series,
Summability and
Resurgence II
Simple and Multiple Summability
Michèle Loday-Richaud
LAREMA, Université d’Angers
Angers, France
ISSN 0075-8434 ISSN 1617-9692 (electronic)
Lecture Notes in Mathematics
ISBN 978-3-319-29074-4 ISBN 978-3-319-29075-1 (eBook)
DOI 10.1007/978-3-319-29075-1
Library of Congress Control Number: 2016940058
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ToHerve´ andSylvie
Avant-Propos
Le sujet principal traite´ dans la se´rie de volumes Divergent Series, Summability
and Resurgence est la the´orie des de´veloppements asymptotiques et des se´ries di-
vergentes applique´e aux e´quations diffe´rentielles ordinaires (EDO) et a` certaines
e´quationsauxdiffe´rencesdanslechampcomplexe.
Lese´quationsdiffe´rentiellesdanslechampcomplexe,etdanslecadreholomor-
phe,sontunsujettre`sancien.Lathe´orieae´te´ tre`sactivedansladeuxie`memoitie´
duXIX-e`mesie`cle.Encequiconcernelese´quationsline´aires,lesmathe´maticiens
de cette e´poque les ont subdivise´es en deux classes. Pour la premie`re, celle des
e´quationsa` pointssinguliersre´guliers(oudeFuchs),ge´ne´ralisantlese´quationshy-
perge´ome´triques d’Euler et de Gauss, ils ont enregistre´ “des succe`s aussi de´cisifs
quefaciles”commel’e´crivaitRene´Garnieren1919.Enrevanche,pourlaseconde,
celledese´quationsditesa` pointssinguliersirre´guliers,commel’e´crivaitaussiGar-
nier,“leurseffortsrestentimpuissantsa` e´difieraucunethe´oriege´ne´rale”.Laraison
centrale de ce vif contraste est que toute se´rie entie`re apparaissant dans l’e´criture
d’unesolutiond’unee´quationdiffe´rentielledeFuchsestautomatiquementconver-
gentetandisquepourlese´quationsirre´gulie`rescesse´riessontge´ne´riquementdiver-
gentes et que l’on ne savait qu’en faire. La situation a commence´ a` changer graˆce
a` un travail magistral de Henri Poincare´ entrepris juste apre`s sa the`se, dans lequel
il “donne un sens” aux solutions divergentes des EDO line´aires irre´gulie`res en in-
troduisant un outil nouveau, et qui e´tait appele´ a` un grand avenir, la the´orie des
de´veloppementsasymptotiques.Ilaensuiteutilise´cetoutilpourdonnerunsensaux
se´ries divergentes de la me´canique ce´leste, et remporte´ de tels succe`s que presque
toutlemondeaoublie´ l’originedel’histoire,c’est-a`-direlesEDO!Lestravauxde
Poincare´ont(unpeu...)remisa`l’honneurl’e´tudedesse´riesdivergentes,abandonne´e
parlesmathe´maticiensapre`sCauchy.L’Acade´miedesSciencesasoumiscesujetau
concoursen1899,cequifuta`l’origined’untravailimportantd’E´mileBorel.Celui-
ciestlasourcedenombredestechniquesutilise´esdansDivergentSeries,Summabil-
ityandResurgence.PourrevenirauxEDOirre´gulie`res,lesujetafaitl’objetdenom-
breuxetimportantstravauxdeG.D.BirkhoffetR.Garnierdurantlepremierquart
duXX-e`mesie`cle.Onretrouveraicidenombreuxprolongementsdesme´thodesde
Birkhoff. Apre`s 1940, le sujet a e´trangement presque disparu, la the´orie e´tant, je
vii
viii Avant-Propos
nesaistroppourquoi,conside´re´ecommeacheve´e,toutcommecelledese´quations
de Fuchs. Ces dernie`res ont re´e´merge´ au de´but des anne´es 1970, avec les travaux
deRaymondGe´rard,puisunlivredePierreDeligne.Lese´quationsirre´gulie`resont
suiviavecdestravauxdel’e´coleallemandeetsurtoutdel’e´colefranc¸aise.Denom-
breuses techniques comple`tement nouvelles ont e´te´ introduites (de´veloppements
asymptotiques Gevrey, k-sommabilite´, multisommabilite´, fonctions re´surgentes...)
permettant en particulier une vaste ge´ne´ralisation du phe´nome`ne de Stokes et sa
miseenrelationaveclathe´oriedeGaloisdiffe´rentielleetleproble`medeRiemann-
Hilbert ge´ne´ralise´. Tout ceci a depuis rec¸u de tre`s nombreuses applications dans
des domaines tre`s varie´s, allant de l’inte´grabilite´ des syste`mes hamiltoniens aux
proble`mes de points tournant pour les EDO singulie`rement perturbe´es ou a` divers
proble`mes de modules. On en trouvera certaines dans Divergent Series, Summa-
bilityandResurgence,commel’e´tudere´surgentedesgermesdediffe´omorphismes
analytiques du plan complexe tangents a` l’identite´ ou celle de l’EDO non-line´aire
Painleve´ I.
Lesujetrestaitaujourd’huidifficiled’acce`s,lelecteurnedisposantpas,misa`part
les articles originaux, de pre´sentation accessible couvrant tous les aspects. Ainsi
Divergent Series, Summability and Resurgence comble une lacune. Ces volumes
pre´sententunlargepanoramadesrechercheslesplusre´centessurunvastedomaine
classique et passionnant, en pleine renaissance, on peut meˆme dire en pleine ex-
plosion.Ilssontne´anmoinsaccessiblesa` toutlecteurposse´dantunebonnefamilia-
rite´ avec les fonctions analytiques d’une variable complexe. Les divers outils sont
soigneusement mis en place, progressivement et avec beaucoup d’exemples. C’est
unebellere´ussite.
A` Toulouse,le16mai2014,
Jean-PierreRamis
Preface to the Three Volumes
Thisthree-volumeseriesaroseoutoflecturenotesforthecourseswegavetogether
ataCIMPA1 schoolinLima,Peru,inJuly2008.Sincethen,thesenoteshavebeen
used and developed in graduate courses held at our respective institutions, that is,
the universities of Angers, Nantes, Strasbourg (France) and the Scuola Normale
SuperiorediPisa(Italy).Theoriginalnoteshavenowgrownintoself-containedin-
troductionstoproblemsraisedbyanalyticcontinuationandthedivergenceofpower
seriesinonecomplexvariable,especiallywhenrelatedtodifferentialequations.
A classical way of solving an analytic differential equation is the power series
method,whichsubstitutesapowerseriesfortheunknownfunctionintheequation,
then identifies the coefficients. Such a series, if convergent, provides an analytic
solution to the equation. This is what happens at an ordinary point, that is, when
we have an initial value problem to which the Cauchy-Lipschitz theorem applies.
Otherwise, at a singular point, even when the method can be applied the resulting
series most often diverges; its connection with “actual” local analytic solutions is
notobviousdespiteitsdeeplinktotheequation.
The hidden meaning of divergent formal solutions was already pondered in the
nineteenthcentury,afterCauchyhadclarifiedthenotionsofconvergenceanddiver-
genceofseries.Forordinarylinear differentialequations,ithasbeenknownsince
thebeginningofthetwentiethcenturyhowtodetermineafullsetoflinearlyinde-
pendentformalsolutions2atasingularpointintermsofafinitenumberofcomplex
powers, logarithms, exponentials and power series, either convergent or divergent.
Theseformalsolutionscompletelydeterminethelineardifferentialequation;hence,
they contain all information about the equation itself, especially about its analytic
solutions.Extractingthisinformationfromthedivergentsolutionswastheunderly-
1 CentreInternationaldeMathe´matiquesPuresetApplique´es,orICPAM,isanon-profitinter-
nationalorganizationfoundedin1978inNice,France.Itpromotesinternationalcooperationin
highereducationandresearchinmathematicsandrelatedsubjectsforthebenefitofdeveloping
countries.ItissupportedbyUNESCOandIMU,andmanynationalmathematicalsocietiesover
theworld.
2 Onesaysaformalfundamentalsolution.
ix
